This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A323626 #15 Feb 27 2020 23:21:28 %S A323626 3,3,1,3,1,2,3,3,1,1,13,1,7,3,1,3,1,2,77,1,1,26,203,1,817,14,109,3, %T A323626 1037,2,3,3,1,1,1297,1,20275,77,155,1,17,1,13,13,67,203,6716227,1, %U A323626 421735,817,17,7,2306997,109,55739,3,49,1037,818712813,1,138203853,3 %N A323626 For any nonnegative real number x, let f(x) be the real number obtained by replacing in the binary expansions of the integer and fractional parts of x each finite run of k consecutive equal bits b with a run of k-(-1)^k consecutive bits b; a(n) is the numerator of f(1/n). %C A323626 When computing f(x), we consider the unique binary representation of x where the fractional part of x does not eventually end with repeating ones. %C A323626 The function f establishes a self-inverse bijection: %C A323626 - over the nonnegative real numbers, %C A323626 - over the nonnegative real numbers in the half-open interval [0,1), %C A323626 - over the nonnegative rational numbers, %C A323626 - over the nonnegative rational numbers in the half-open interval [0,1), %C A323626 - over the nonnegative integers (for any n >= 0, f(n) = A162853(n)). %C A323626 The function f has only one fixed point: f(0) = 0. %H A323626 Rémy Sigrist, <a href="/A323626/a323626.png">Representation of f in the half-open interval [0,1)</a> %H A323626 Rémy Sigrist, <a href="/A323626/a323626_1.gp.txt">PARI program for A323626</a> %F A323626 a(2^k) = 3 for any k >= 0. %F A323626 a(2^k-1) = 2-(-1)^k for any k > 0. %e A323626 The first terms of the sequence, alongside f(1/n) and the binary representations of 1/n and of f(1/n) with periodic part in parentheses, are: %e A323626 n a(n) f(1/n) bin(1/n) bin(f(1/n)) %e A323626 -- ---- ------- ---------------------- ------------------------ %e A323626 1 3 3 1.(0) 11.(0) %e A323626 2 3 3/4 0.1(0) 0.11(0) %e A323626 3 1 1/5 0.(01) 0.(0011) %e A323626 4 3 3/16 0.01(0) 0.0011(0) %e A323626 5 1 1/3 0.(0011) 0.(01) %e A323626 6 2 2/5 0.0(01) 0.(0110) %e A323626 7 3 3/7 0.(001) 0.(011) %e A323626 8 3 3/8 0.001(0) 0.011(0) %e A323626 9 1 1/17 0.(000111) 0.(00001111) %e A323626 10 1 1/24 0.0(0011) 0.000(01) %e A323626 11 13 13/257 0.(0001011101) 0.(0000110011110011) %e A323626 12 1 1/20 0.00(01) 0.00(0011) %e A323626 13 7 7/129 0.(000100111011) 0.(00001101111001) %e A323626 14 3 3/56 0.0(001) 0.000(011) %e A323626 15 1 1/21 0.(0001) 0.(000011) %e A323626 16 3 3/64 0.0001(0) 0.000011(0) %e A323626 17 1 1/9 0.(00001111) 0.(000111) %e A323626 18 2 2/17 0.0(000111) 0.(00011110) %e A323626 19 77 77/1025 0.(000011010111100101) 0.(00010011001110110011) %e A323626 20 1 1/12 0.00(0011) 0.00(01) %o A323626 (PARI) See Links section. %Y A323626 See A323627 for the corresponding denominators. %Y A323626 Cf. A162853. %K A323626 nonn,frac,base %O A323626 1,1 %A A323626 _Rémy Sigrist_, Jan 20 2019